Correlation-Based Radar Receivers with Pulse-Shaped OFDM Signals

an author's https://oatao.univ-toulouse.fr/26532

Mercier, Steven and Roque, Damien and Bidon, Stéphanie and Enderli, Cyrille Correlation-Based Radar Receivers with Pulse-Shaped OFDM Signals. (2020) In: Radar Conference 2020, 21 September 2020 - 25 September 2020 (Florence, Italy).

Correlation-Based Radar Receivers with

Pulse-Shaped OFDM Signals

Steven Mercier

, Damien Roque

, St´ephanie Bidon

and Cyrille Enderli

∗_{ISAE-SUPAERO, Universit´e de Toulouse, France}

Email: {steven.mercier,stephanie.bidon,damien.roque}@isae-supaero.fr

†_{Thales DMS, ´Elancourt, France}

Email: cyrille-jean.enderli@fr.thalesgroup.com

Abstract—In waveform sharing scenarios, various radar re-ceivers have been developed for orthogonal frequency-division multiplexing (OFDM) signals. More general waveforms, such as pulse-shaped multicarrier modulations received little attention so far, despite their increased robustness to high-Doppler scatterers. In this paper, we compare the performance of two correlation-based radar receivers, namely the matched filter and the symbol-based technique, when used with different pulse-shaped multicar-rier waveforms. We express the signal-to-interference-plus-noise-ratio in the range-Doppler map, taking into account the pedestal (or random sidelobes) induced by the symbols. Benefits of pulse shaping is further illustrated in a realistic vehicular scenario, in presence of multiple targets and ground clutter. In this context, the symbol-based approach outperforms the matched filter while enjoying a low-computational complexity.

More generally, our results reveal the multicarrier pulse shape as a relevant degree of freedom in waveform co-design approaches (e.g., cognitive radar/communication systems).

I. INTRODUCTION

Dual-functional radar-communication (DFRC) systems are intended to decongest the radio spectrum while sharing a single hardware platform [1]. They rely on a joint waveform design to simultaneously perform both functionalities (Fig. 1). In various applications such as vehicular networks, multicar-rier modulations are usually good candidates to accommo-date highly time-frequency selective radar and communication channels [2].

From a correlation-based radar receiver viewpoint, the sym-bols carried by the multicarrier waveform can be seen as a nuisance parameter, inducing a pedestal (or random sidelobes) in the range-Doppler map. While such an interference has been studied in the particular case of cyclic-prefix orthogonal frequency-division multiplexing (CP-OFDM) [3], a general-ization to pulse-shaped multicarrier schemes (e.g., FBMC1_,

FMT2 _{[4]) is of interest to increase the robustness of DFRC}

systems towards high-Doppler and/or low signal-to-noise-ratio scenarios [5], [6].

In this paper, we quantify the impact of that limitation in the context of weighted cyclic prefix (WCP)-OFDM wave-forms [7], which are an extension of the conventional CP-OFDM to possibly non-rectangular pulse-shapes while

pre-This work has been supported by DGA/MRIS under grant 2017.60.0005 and Thales DMS.

1_{Filter Bank based MultiCarrier.} 2_{Filtered MultiTone.} r[p] Shared TX Radar RX {ck,m} DFRC transceiver e.g., vehicle, base-station

Radar channel Communication channel

Com. RX e.g., vehicle, base-station s[p]

Fig. 1. Vehicular DFRC scenario involving a shared waveform to simultane-ously perform radar sensing and data transmission. The radar transceiver is monostatic with a perfect knowledge of the symbol sequence.

serving a low-complexity implementation. Two remarkable correlation-based radar receivers are especially considered: the well-known matched filter (MF), and the so-called symbol-based (SB) processing [8]. We first compare their signal-to-interference-plus-noise-ratio (SINR) in response to a single-scatterer scenario, for both time-frequency localized (TFL) and rectangular pulse-shaped transmissions (the latter being the CP-OFDM baseline). The benefits of pulse-shaping is then further exemplified in a simulated realistic monostatic vehicular scenario, including several targets and ground clutter. Notation: IN denotes the finite set {0, . . . , N − 1} and \

the set difference. E {·} is the expectation operator, k·k the `2-norm and sgn is the signum function anywhere but in 0

where we set sgn(0)_{, 1.}

II. MULTICARRIER RADAR SYSTEM

In this Section, we describe the radar “path” depicted in Fig. 1 (thick arrows). A detailed performance study of the communication link falls beyond the scope of this paper. A. Shared multicarrier transmitter

The DFRC transmitter generates M blocks (or sweeps) of K orthogonal subcarriers over an instantaneous bandwidth B around a carrier frequency Fc. Each block/subcarrier carries a

complex data symbol ck,mwith (k, m) ∈ IK×IM. Assuming

K  1, the critically sampled transmitted signal is [9] s[p] , M −1 X m=0 1 √ K K−1 X k=0 ck,mej2π k K(p−mL) ! g[p − mL] (1)

where g is the transmit pulse-shape and L ≥ K is the discrete-time pulse repetition interval. In the following, symbols are assumed independent and uniformly distributed, drawn from a zero-mean unit-variance constellation of size N . The raw spectral efficiency of the system is thus η = log2(N )K/L. To

ensure a low-complexity implementation of (1) in the time-domain, we focus on short pulses (i.e., g[p] = 0 for p /∈ IL [7]).

B. Radar channel

Target: While it propagates towards the communication receiver at light speed c, part of the narrowband signal (1) is backscattered to the DFRC transceiver by a target. The latter is described as a single point object with: a zero-mean complex amplitude α; a radial velocity v and an attendant Doppler frequency Fd , 2vFc/c; an unambiguous range R0 = l0δR

with l0 ∈ IK the range gate and δR , c/(2B) the radar

range resolution. Provided that Fd  B, the baseband target

signal at the input of the radar receiver is sampled at rate B to yield [9]

rt[p] = α exp(j2πFdp/B)s[p − l0]. (2)

Ground clutter: Presence of ground echoes in the re-ceived radar signal may also be expected. We model ground clutter as a contiguous set of statistically independent patches in range and azimuth. Especially, at each range bin ic ∈ Jc

with Jc , IK\Ibhr/δRcwhere hrdenotes the system altitude

and b·c the floor function, we assume Nc equally spaced

patches around the radar. As for the target, each patch is characterized by: a zero-mean complex amplitude ρic,nc; a

nominal Doppler frequency Fd_ic,nc  B. Consequently, the

received baseband clutter signal sampled at rate B is given by rc[p] =

X

ic∈Jc

X

nc∈INc

ρic,ncexp(j2πFdic,ncp/B)s[p−ic]. (3)

Apart from the realistic scenario under consideration in Sec-tion IV, we will presume in the following that rc[p] = 0.

Additive thermal noise: A thermal noise contribution w, modeled as a zero-mean white circular Gaussian process with variance σ2_{, finally adds up to (2) and (3), yielding the full}

received baseband radar signal r[p] =

(

rt[p] + rc[p] + w[p] if p ∈ ILM,

0 otherwise. (4)

Truncation to LM samples in (4) enables a coherent process-ing interval (CPI) restricted to the transmitted signal length, thus preserving compact notation in the upcoming derivations. In most realistic scenarios (e.g., M  1) a negligible integra-tion loss is reported.

C. Correlation-based radar receivers

With the aim of detecting and estimating the target, (4) is usually represented in the range-Doppler domain3_{. Among the}

3_{In numerous scenarios, such transformation is preceded by a clutter}

mitigation stage (e.g., whitening). The latter is however out of the scope of this work.

variety of correlation-based receivers that have accordingly been proposed, two stand out [3]. These are recalled here-after with a normalization factor chosen to ensure the same processing gain for a target at null range and Doppler.

Matched filter (MF): The matched filter is a classical radar receiver that consists in cross-correlating the received signal (4) with Doppler shifted versions of the transmitted signal (1). It can be written

χMF[l, n] , 1 √ KM LM −1 X p=0 r[p]s∗[p − l]e−j2πM Ln p. (5)

While it is known to be the optimum linear filter in terms of signal-to-noise-ratio (SNR) in presence of white noise, such a performance criterion is not necessarily adequate for information-bearing signals, as discussed hereafter in Subsec-tion II-D.

Symbol-based (SB): The so-called symbol-based radar

receiver can be seen as a simplification of the MF relying on (i) an invariant Doppler approximation for each block, (ii) the exploitation of the multicarrier waveform structure, revealing a blockwise FFT-based implementation [3]. It basically consists of the three following stages [6], [8].

1) Estimation of the data symbols via a regular linear multicarrier receiver: ˜ ck0_,m0 , 1 √ K +∞ X p=−∞ r[p]ˇg∗[p − m0L]e−j2πk0K(p−m 0_L) (6a) with ˇg the receive pulse-shape verifying the perfect reconstruction condition [7].

2) Channel estimation via data symbol removal: ¯ ck0_,m0 , ˜ ck0_,m0 ck0_,m0 = (a)˜ck 0_,m0c∗_k0_,m0. (6b)

Note (a): the identity is obtained by considering PSK modulations, as assumed in the rest of this paper. 3) Range-Doppler map computation via an IDFT/DFT pair:

χSB[l, n] , 1 √ KM M −1 X m0₌₀ K−1 X k0₌₀ ¯ ck0_,m0ej2π k0 Kl_e−j2πm0Mn_. (6c) D. Range-Doppler maps tainted with random sidelobes

As proved in [3], range-Doppler maps resulting from the MF (5) and SB (6) receivers in a clutter-free scenario can be expressed as (7) and (8), respectively. w_MF0 and w0_SB denote the output thermal noise signals and

A(g,g)_{(l, f ) ,} 1 K L−1 X p=0 g[p]g∗[p − l]ej2πf p (9) A(g,ˇg)_{(l, f ) ,} 1 K L−1 X p=0 g[p]ˇg∗[p − l]ej2πf p (10)

are the pulse-ambiguity functions. These range-Doppler maps are especially tainted with so-called random sidelobes (or

χMF[l, n] ' M 1αe j2πFd_B− n M L  l0X k,m X k0_,m0 ck,mc ∗ k0_,m0Ψk0_,m,m0_,l,nA(g,g)  l − l0+ (m 0 − m)L,Fd B − n M L+ k − k0 K  + w0MF[l, n] (7) χSB[l, n] ' M 1αe j2πFd_Bl0X k,m X k0_,m0 ck,m ck0_,m0e j2πn M(m−m 0₎ Ψk0_,m,m0_,l,nA(g,ˇg)  −l0+ (m0− m)L, Fd B + k − k0 K  + w0SB[l, n] (8) with Ψk0_,m,m0_,l,n= ej2π k0 K(l−l0+(m0−m)L)_ej2π _Fd BL−Mn  m /√KM

pedestal). This interference can actually be inferred from (7)– (8) by considering any observation cell distinct from the target’s (i.e., [l, n] 6= [l0, FdM L/B]). Aside from the thermal

noise contribution, the double sum involves the transmitted

random data symbols ck,m along with the pulse-ambiguity

functions (9)–(10) evaluated at coordinates where they may have non-zero values. As a hint, their cuts are displayed in Fig. 2 for the two pairs of perfect reconstruction pulse-shapes that will be of interest in the rest of this paper, namely:

• the rectangular, or cyclic prefix (CP) pulses, leading to the conventional CP-OFDM waveform:

gCP[p] = (p K/L if p ∈ IL 0 otherwise ˇ gCP[p] = (p L/K if p ∈ IL\ IL−K 0 otherwise

where L − K defines the cyclic prefix length.

• the time-frequency localized (TFL) pulses:

gTFL[p] = ˇgTFL[p] =          cos θ[p] if p ∈ IL−K 1 if p ∈ IK\ IL−K sin θ[p] if p ∈ IL\ IK 0 otherwise

where {θ[p]}L−K−1_p=0 is given in [10] through numerical optimization.

Particularly, in the remainder of this paper, the influence of these pulse-shapes on the level of random sidelobes is examined.

III. THEORETICALSINRPERFORMANCE

The impact of pulse-shaping is first assessed on the signal-to-interference-plus-noise-ratio (SINR) metrics in a clutter-free single-target scenario. This constitutes an indication regarding detection performance. Without loss of generality, we also assume in this Section that the target’s Doppler frequency verifies

Fd

BL ,

n0

M with n0∈ IM.

Considering a range-Doppler map χ, we define the output SINR as SINRχ[l, n] , P_t(χ)[l0, n0] P_i(χ)[l, n] + P_w(χ)0 , [l, n] 6= [l0, n0] (11) −L −(L − K) 0 L 0 0.2 0.4 0.6 0.8 1 l Amplitude |A(g,g)_CP (l, f )| |A(g,ˇ_CPg)(l, f )| |A(g,g)_TFL (l, f )| = |A(g,ˇ_TFLg)(l, f )|

(a) Zero-Doppler cut (i.e., f = 0)

0 1 L 1 K 0 0.2 0.4 0.6 0.8 1 f Amplitude

(b) Zero-range cut (i.e., l = 0)

Fig. 2. Pulse-ambiguity functions cuts for CP and TFL pulses (L/K = 12/8)

where Pt(χ)[l0, n0], P (χ)

i [l, n] and P

(χ)

w0 denote the target

peak, target’s random sidelobes and output noise powers, respectively.

In our case, the SINR expressions of the MF and SB re-ceivers were shown to result in (12)–(13) [3]. It is worth noting that unlike the MF, in a given scenario, the SB receiver has a uniform SINR in the range-Doppler map. In any case, both SINRs depend on the target parameters [l0, n0] and, naturally,

on the pulse-shapes via their pulse-ambiguity functions. As such, we represent in Figs. 3–5 different cuts of these SINRs, considering varying target positions in the range-Doppler map, for both CP and TFL pulses. We chose K = 1024,

SINRχMF[l, n] ' M 1 KM P k6=0  A(g,g) _{l − l} 0,n_LM0−n+_Kk
  2 +P k  A(g,g) _{l − l} 0− sgn(l − l0)L,n_LM0−n+_Kk
  2 + σ2 E{|α|2_} (12) SINRχSB[l, n]_{M 1}' KM   A (g,ˇg) _−l 0,_LMn0
   2 P k6=0  A(g,ˇg) _−l 0,_LMn0 +_Kk
 2 +P k  A(g,ˇg) _−l 0+ L,_LMn0 +_Kk
 2 +kˇgk_K2 σ2 E{|α|2_} (13) 0 200 400 600 800 1,000 49 50 51 Range bin SINR (dB) MF-CP MF-TFL SB-CP SB-TFL (a) Cut at n = 1 0 200 400 600 800 1,000 49 50 51 Range bin SINR (dB) MF-CP MF-TFL SB-CP SB-TFL (b) Cut at n = M/2 Fig. 3. SINR Doppler cuts for target at [l0, n0] = [(L − K)/2, 0] = [64, 0] (i.e., static target, prior to the CP)

0 200 400 600 800 1,000 49 50 51 Range bin SINR (dB) (a) Cut at n = 1 0 200 400 600 800 1,000 49 50 51 Range bin SINR (dB) (b) Cut at n = M/2

Fig. 4. SINR Doppler cuts for target at [l0, n0] = [(L − K)/2, M/4] = [64, 32] (i.e., moving target, prior to the CP)

0 200 400 600 800 1,000 42 44 46 48 50 Range bin SINR (dB) (a) Cut at n = 1 0 200 400 600 800 1,000 42 44 46 48 50 Range bin SINR (dB) (b) Cut at n = M/2 Fig. 5. SINR Doppler cuts for target at [l0, n0] = [L/2, M/4] = [576, 32] (i.e., moving target, exceeding the CP)

As expected from (12)–(13) and from the pulse-ambiguity cuts (see the solid lines in Fig. 2), we observe in these figures that the SINR of the MF decreases as the distance between the observation bin and that of the target grows. As a result, although it is always locally higher near the target, it often gets outperformed by the uniform SINR of the SB receiver farther in the map, as evidenced by Figs. 3 and 4. An exception occurs when the target’s range or/and velocity becomes significant as seen in Fig. 5. Indeed, in such a case, the target peak’s integration lossA(g,ˇg)_(−l

0, n0/LM )

2

incurred by the SB (13) highly penalizes the SINR (see the blue solid and orange dashdotted lines in Fig. 2).

With the SB receiver, we can notice from Figs. 3 and 4 that a higher SINR is achieved by the TFL pulses, as compared to the CP pulses, when the target has a low range and/or high velocity, thereby confirming the observations of [9]. In these situations, CP pulses indeed suffer from their increased noise power by factor L/K and from a more significant pedestal (see the higher sidelobes of the orange dashdotted line as compared to the blue one in Fig. 2b). This gets compensated for a sufficiently distant target l0 as seen in Fig. 5. Finally, for the

MF receiver, we observe only slight improvements brought by the TFL in these figures, in particular nearby the target location where the random sidelobes power approaches 0, unlike with the CP (see P(χMF)

i [l0, n0] in (12) together with Fig. 2). They

can be more significant with higher values of L/K though, since TFL pulses then offer better time-frequency localization (not represented here for the sake of compactness).

Whereas the SINR differences discussed in this Section are not always significant, we recall that they are observed when considering the pedestal induced by one single target only. Their repercussions on detection could be more severe in a realistic scenario, as hinted hereafter.

IV. AUTOMOTIVE SCENARIO

We now propose to examine the influence of pulse-shaping regarding the random sidelobes interference in a typical real-world application, namely automotive DFRC. The system parameters used for the simulation are inspired from the practical implementation proposed in [11, Table 3]. As such, the DFRC system operates in the ISM band, at Fc= 24 GHz,

occupies a bandwidth B = 93.1 MHz split into K = 1024 subcarriers, over a duration of M = 128 blocks. The transmit

power is Ps = −32 dBW. We set L/K = 9/8 according to

the power delay profile of the communication channel (see Fig. 1) while preserving a good-enough spectral efficiency.

The vehicle equipped with the DFRC system is assumed to be on the left lane of a highway, driving straight at constant velocity vr= 31.5 m/s. The transmit and receive antennas are

identical uniform rectangular arrays of 8 × 6 elements having cosine patterns but no backlobes, disposed on the front car bumper, that is at altitude hr= 0.5 m from the ground. They

achieve gains GRX= GTX' 22 dBi in the vehicle’s direction.

The radar scene is comprised of 6 target vehicles: 4 driving in our direction, 2 in the opposite one. Their parameters (namely range, radial velocity and radar cross sections) are

TABLE I TARGETS PARAMETERS

Target Motorcycle Car 1 Truck Car 2 Car 3 Car 4

Range (m) 80.6 177.2 30.6 96.7 41.9 122.4

Rad. vel. (m/s) −4 0 8 4 62.7 63.1

RCS (dBsm) 5 14 35 14 12 12

provided in Table I. It also includes some ground clut-ter. The latter is generated following (3) with Nc = 360

while assuming a Gaussian distribution of the patches, i.e., ρic,nc ∼ CN (0, E|ρic,nc|

2 _{). Their powers E |ρ} ic,nc|

2 are computed from the radar equation for area clutter using the constant gamma reflectivity model [12] with γ = −5 dB. Ultimately, the input white Gaussian noise w is simulated with σ2_{= −118 dBW. The range-Doppler maps resulting from the}

sum channel of the receive antenna array and focused on the portion of interest are depicted in Fig. 6.

Due to the cumulation of the scatterers random sidelobes, the average background levels appear at least 6 dB higher than the expected post-processing noise powers. Regardless of the pulse-shape, we remark a substantially lower excess at the output of the SB as compared to the MF. Concomitantly, the measured target peak powers remain comparable from one receiver to the other. This is in line with the observations drawn in Section III for the near and/or fast targets (Figs. 3 and 4) as found in this automotive scenario. For both receivers, we also perceive improvements when using TFL pulses as compared to CP. Unlike in Section III, this is particularly visible with the MF here. Either way, these results emphasize the relevance of pulse-shaping to preserve the dynamic range even in the context of DFRC systems.

V. CONCLUSION AND FUTURE WORK

In this paper, we investigated the performance of two correlation-based radar receivers, commonly encountered in OFDM DFRC, while extending them to the more general context of pulse-shaped multicarrier waveforms. Assessment was conducted for two short pulses, namely the conventional cyclic prefix baseline and the time-frequency localized pulses. We focused our analysis on the interference level related to the symbols-induced random sidelobes. Both the derived SINR expressions and the synthesized realistic automotive scenario have exhibited the interest of TFL pulses regarding inter-ference in highly-mobile short-range scenarios, particularly when used together with the so-called symbol-based receiver. More broadly, these results tend to show the possible benefits brought by pulse-shaping to DFRC systems. Future work may include an extension of this study to alternate pulse-shaped multicarrier waveforms.

−79.2 dBW −84.6 dBW −38.0 dBW −74.2 dBW −67.6 dBW −80.9 dBW −20 0 20 40 60 80 0 50 100 150 200 Radial velocity (m/s) Range (m) −120 −100 −80 −60 (dB) (a) Matched filter with CP pulses. Measured average background power: −97.7 dBW. −79.4 dBW −84.5 dBW −38.0 dBW −74.2 dBW −68.0 dBW −80.8 dBW −20 0 20 40 60 80 0 50 100 150 200 Radial velocity (m/s) Range (m) −120 −100 −80 −60 (dB) (b) Symbol-based with CP pulses. Measured average background power: −111.7 dBW. −79.5 dBW −84.4 dBW −38.0 dBW −74.2 dBW −67.7 dBW −80.1 dBW −20 0 20 40 60 80 0 50 100 150 200 Radial velocity (m/s) Range (m) −120 −100 −80 −60 (dB) (c) Matched filter with TFL pulses. Measured average background power: −102.9 dBW. −79.7 dBW −85.2 dBW −38.1 dBW −74.4 dBW −68.0 dBW −81.0 dBW −20 0 20 40 60 80 0 50 100 150 200 Radial velocity (m/s) Range (m) −120 −100 −80 −60 (dB) (d) Symbol-based with TFL pulses. Measured average background power: −112.4 dBW.

Fig. 6. Range-Doppler maps of the automotive scenario focused on the portion of interest (Doppler dimension oversampled by a factor 4)

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